On solvability of a nonlinear system of equations for the Fourier-Chebyshev coefficients in the problem of solving ordinary differential equations using Chebyshev series
Keywords:ordinary differential equations, approximate analytical methods, numerical methods, orthogonal expansions, shifted Chebyshev series, Markov’s quadrature formulas
A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier-Chebyshev coefficients is formulated and proved. This theorem is a theoretical substantiation for the previously proposed numerical-analytical method of solving ordinary differential equations using Chebyshev series.
- O. B. Arushanyan and S. F. Zaletkin, “Approximate Solution of the Cauchy Problem for Ordinary Differential Equations by the Method of Chebyshev Series,” Vychisl. Metody Programm. 17, 121-131 (2016).
- O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Calculation of Expansion Coefficients of Series in Chebyshev Polynomials for a Solution to a Cauchy Problem,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 5, 24-30 (2012) [Moscow Univ. Math. Bull. 67 (5-6), 211-216 (2012)].
- I. S. Berezin and N. P. Zhidkov, Computing Methods (Fizmatgiz, Moscow, 1962; Pergamon, Oxford, 1965).
- E. Hairer, S. P. Nörsett, and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems (Springer, Berlin, 1987; Mir, Moscow, 1990).
- R. England, “Error Estimates for Runge-Kutta Type Solutions to Systems of Ordinary Differential Equations,” Comput. J. 12 (2), 166-170 (1969).
- C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice Hall, Englewood Cliffs, 1971).
- O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Application of Chebyshev Series to Integration of Ordinary Differential Equations with Rapidly Growing Solutions,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 5, 57-60 (2015) [Moscow Univ. Math. Bull. 70 (5), 237-240 (2015)].
- O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Approximate Solution of Ordinary Differential Equations Using Chebyshev Series,” Sib. Elektron. Mat. Izv. 7, 122-131 (2010).
How to Cite
Copyright (c) 2017 Вычислительные методы и программирование
This work is licensed under a Creative Commons Attribution 4.0 International License.